8 Bicomplexes

Each bicomplex in homalg has an underlying complex of complexes. The bicomplex structure is simply the addition of the known sign trick which induces the obvious equivalence between the category of bicomplexes and the category of complexes with complexes as objects and chain morphisms as morphisms. The majority of filtered complexes in algebra and geometry (unlike topology) arise as the total complex of a bicomplex. Hence, most spectral sequences in algebra are spectral sequences of bicomplexes. Indeed, bicomplexes in homalg are mainly used as an input for the spectral sequence machinery.

8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep

(It is a representation of the GAP category IsHomalgBicomplex (8.1-1), which is a subrepresentation of the GAP representation IsFinitelyPresentedObjectRep.)

8.2 Bicomplexes: Constructors

8.2-1 HomalgBicomplex

‣ HomalgBicomplex( C )

( function )

Returns: a homalg bicomplex

This constructor creates a bicomplex (homological bicomplex) given a homalg complex of (co)complexes C (--> HomalgComplex (6.2-1)), resp. creates a bicocomplex (cohomological bicomplex) given a homalg cocomplex of (co)complexes C (--> HomalgCocomplex (6.2-2)). Using the usual sign-trick a complex of complexes gives rise to a bicomplex and vice versa.